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Properties

Label 2-261-9.4-c1-0-8
Degree $2$
Conductor $261$
Sign $-0.870 - 0.492i$
Analytic cond. $2.08409$
Root an. cond. $1.44363$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.973 + 1.68i)2-s + (0.738 + 1.56i)3-s + (−0.896 + 1.55i)4-s + (−1.26 + 2.19i)5-s + (−1.92 + 2.77i)6-s + (−2.23 − 3.87i)7-s + 0.404·8-s + (−1.90 + 2.31i)9-s − 4.94·10-s + (1.89 + 3.27i)11-s + (−3.09 − 0.257i)12-s + (1.42 − 2.47i)13-s + (4.35 − 7.54i)14-s + (−4.38 − 0.365i)15-s + (2.18 + 3.78i)16-s − 1.09·17-s + ⋯
L(s)  = 1  + (0.688 + 1.19i)2-s + (0.426 + 0.904i)3-s + (−0.448 + 0.776i)4-s + (−0.567 + 0.983i)5-s + (−0.785 + 1.13i)6-s + (−0.845 − 1.46i)7-s + 0.143·8-s + (−0.636 + 0.771i)9-s − 1.56·10-s + (0.570 + 0.988i)11-s + (−0.892 − 0.0744i)12-s + (0.395 − 0.685i)13-s + (1.16 − 2.01i)14-s + (−1.13 − 0.0943i)15-s + (0.546 + 0.946i)16-s − 0.264·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $-0.870 - 0.492i$
Analytic conductor: \(2.08409\)
Root analytic conductor: \(1.44363\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 261,\ (\ :1/2),\ -0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.458463 + 1.73951i\)
\(L(\frac12)\) \(\approx\) \(0.458463 + 1.73951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.738 - 1.56i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.973 - 1.68i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.26 - 2.19i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.23 + 3.87i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.89 - 3.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.42 + 2.47i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.09T + 17T^{2} \)
19 \( 1 - 5.81T + 19T^{2} \)
23 \( 1 + (-2.18 + 3.78i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (1.12 - 1.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.17T + 37T^{2} \)
41 \( 1 + (0.695 - 1.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.95 + 8.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.95 + 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.52T + 53T^{2} \)
59 \( 1 + (-3.27 + 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.46 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.01 - 5.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.67T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + (-7.22 - 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.05 - 7.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (7.45 + 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.76142501387050973191942118409, −11.15992638528944686630218582379, −10.41085319736794331930312880088, −9.660913589576517963542549396132, −8.100119485082547812577699883939, −7.15712699559520369363120873000, −6.68390907377724018337561494601, −5.13583619387562085835090429560, −3.99033469611770512136467842486, −3.35407551147320476387661970617, 1.29531074597121763761063862224, 2.81790200132871674456056663197, 3.69453986475852726907303584405, 5.27005087588886231884281304646, 6.35126759703222794514315403513, 7.85153290637060933500295961946, 8.965576030713452224178795567442, 9.392375944571087838355557623792, 11.34912551916271898992616035118, 11.82270261053236360581385118959

Graph of the $Z$-function along the critical line